One of the most important objects in spectral geometry is the eigenvalue counting function, say, for the Dirichlet Laplacian associated with planar domains. The simplest examples of such domains might be squares, disks, annuli, etc. It is well-known that for each of these domains its eigenvalue counting function has an asymptotics containing two main terms and a remainder of size $o(/mu)$. (Such an asymptotics is usually called Weyl's law.) To improve the estimate of the remainder term had been one of the most attractive problems in spectral geometry for decades.

      In this talk I will introduce background and explain how to transfer the above problem into problems of counting lattice points, to which tools from analysis and analytic number theory can be applied. In particular I will mention our recent work for planar annuli and balls in high dimensions, joint with Wolfgang Mueller, Weiwei Wang and Zuoqin Wang.

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